View source for Abel differential equation

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The ordinary differential equation 
$$
y' = f_0(x) + f_1(x)y + f_2(x)y^2 + f_3(x)y^3
$$
(Abel's differential equation of the first kind) or 
$$
\left(g_0(x) + g_1(x)y \right)y' = f_0(x) + f_1(x)y + f_2(x)y^2 + f_3(x)y^3
$$
(Abel's differential equation of the second kind). These equations arose in the context of the studies of N.H. Abel {{Cite|Ab}} on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural
generalization of the [[Riccati equation|Riccati equation]].

If $f_1 \in C(a,b)$ and $f_2,f_3 \in C^1(a,b)$ and $f_3(x) \neq 0$ for $x \in [a,b]$, then Abel's differential equation of the first kind can be reduced to the normal form $\mathrm{d}z/\mathrm{d}t = z^3 + \Phi(t)$ by substitution of variables {{Cite|Ka}}. In the general case, Abel's differential equation of the first kind cannot be integrated in closed form, though this is possible in special cases {{Cite|Ka}}. If $g_0,g_1 \in C^1(a,b)$ and $g_1(x) \neq 0$, $g_0(x) + g_1(x)y \neq 0$, Abel's differential equation of the second kind can be reduced to Abel's differential equation of the first kind by substituting $g_0(x) + g_1(x)y = 1/z$.

Abel's differential equations of the first and second kinds, as well as their further generalizations 
$$
y' = \sum_{i=0}^n f_i(x)y^i, \quad
y' \sum_{j=0}^m g_j(x)y^j = \sum_{i=0}^n f_i(x)y^i,
$$
have been studied in detail in the complex domain (see, for example, {{Cite|Go}}).

====References====
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|valign="top"|{{Ref|Ab}}||valign="top"| N.H. Abel, "Précis d'une théorie des fonctions elliptiques" ''J. Reine Angew. Math.'', '''4''' (1829) pp. 309–348
|-
|valign="top"|{{Ref|Go}}||valign="top"| V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft.  (1958) (Translated from Russian)
|-
|valign="top"|{{Ref|Ka}}||valign="top"| E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint (1971)
|-
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